Local-to-Global Algorithms in Biology: Drosophila and Beyond (maybe...)

Transcription

1 Local-to-Global Algorithms in Biology: Drosophila and Beyond (maybe...) MIT Synthetic Biology Working Group Dan Yamins

2 spatial multi-agent systems a space with agents embedded in the space local information and processing globally defined tasks

3 engineering: Sensor Networks McLurkin irobot Swarm Saul Griffith s selffolding structures Butera s paintable computer concept

4 biology: drosophila embryo What does thinking of natural spatial multi-agent systems qua computers tell us scientifically? What agent resource capacities are required to solve a given task? Can we then turn around this knowledge to build a better bug?

5 outline. Drosophila background II. Multi-agent systems Model of Drosophila III. An information bound IV. The Radius/State Tradeoff The first four sections study a known system, showing how multi-agent systems reasoning can help us understand features of local-to-global systems design. V. Designing Local Rules The fifth section takes this knowledge, and applies it to develop the beginnings of a protocol for designing novel systems.

6 Drosophila Background (Actually... one stage in early Drosophila, utterly simplified)

7 Drosophila Drosophila melanogaster embryo about minutes post-fertilization

8 Drosophila Let s think of the cell as a... PROTEIN mrna... bag of biomolecules en MID HIghlighted here are 3 substances that play are involved in the stripes in the picture on the previous slide mid CN slp wg hh ci HH

9 Considering cells along the A-P axis: en en en en en en MID MID MID MID MID MID slp hh slp hh slp hh slp hh slp hh slp hh mid wg HH mid wg HH mid wg HH mid wg HH mid wg HH mid wg HH CN ci CN ci CN ci CN ci CN ci CN ci... there is a periodic pattern in the concentration of each of various substances depicted on the previous slide: WinGless grailed SLoPpy Mid HedgeHog Cubitus Interruptus repressor Cubitus Interruptus (protein) Cubitus Interruptus (mrna) The concentration profile pattern at the molecular level corresponds the observed stripe pattern.

10 Drosophila PROTEIN mrna en en slp hh cell Positive Induction mid slp wg MID hh HH mid MID wg HH Negative Inhibition CN The bag of biomolecules... ci CN ci... is actually a Gene Regulatory Network Basal Production

11 en cell mid slp MID wg hh HH Some components of the Gene Regulatory Network in one cell... CN ci cell en cell cell... influence those in neighboring cells. HH mid slp MID wg hh HH HH CN ci

12 It s a spatially coupled Gene Regulatory Network ds i j dt Such a network can be modeled as a coupled ODE: = f j ( S i, S i 2,..., S i N, S i, S i And the ODE will (a la Garrett Odell) have the pattern as a stable nondegenerate steady state: 2,..., S i N, Si+, S2 i+,... S i+ ) N WinGless grailed SLoPpy Mid HedgeHog Cubitus Interruptus repressor Cubitus Interruptus (protein) Cubitus Interruptus (mrna)

13 a multi-agent model of drosophila (a sketchy multi-agent model of drosophila)

14 a simple -D organism, each cell of which has some internal state (,,2,3, etc...)

15 multi-agent model local rules by which the agent integrates information about states of nearby agents and takes a differentiation action action of local rule, on radius-2 ball, changing --> any radius- (nearest neighbor) local rule is allowed.

16 multi-agent model size 4 instance size 8 instance size 2 instance grailed

17 multi-agent model Add slots. Agent with unstructured internal state. Structured multi-register internal states. F( ) Multislot dynamics

18 multi-agent model MID HH CN ci WinGless grailed SLoPpy Mid HedgeHog Cubitus Interruptus repressor Cubitus Interruptus (protein) Cubitus Interruptus (mrna)

19 F( ) en wg ptc cid D CN PH hh PTC HH multi-agent model the gene regulatory network as a nearest-neighbor local rule the brains of the cellular-agent

20 An informational bound

21 Local Checkability Consider the repeat pattern:.... Can this pattern be solved robustly with a nearest-neighbor rule? Answer: No. Because the with a radius rule, would have to be a fixed state.....

22 local checkability; multi-agent reasoning about local equilibria A function Θ : B r {, } is a local check scheme for a pattern T if (Θ(B r (i, X)) = ) X T i X This means: if all cells think criterion Theta is true... then actually the organismal pattern is correct.... has a radius-2 check scheme. All repeated patterns are locally checkable. Any pattern thus has a minimal check radius, a lower bound on how far local rules have to see to form the pattern robustly.

23 local checkability; multi-agent reasoning about local equilibria Hey. Wait. grailed * Local check radius of this pattern is = 2, and in fact to actually make a rule radius 5/2 is required. But the coupled gene network rules are nearest neighbor (i.e. r = ). Put another way, no equation of the form: ds i j dt = f j ( S i, S i 2,..., S i N, S i, S i 2,..., S i N, Si+, S2 i+,... S i+ ) N can have a nondegenerate stable steady state of the form * if engrailed is decoupled. The system is trying to make a 4-coordinate...

24 The Radius/State Tradeoff

25 The Radius/State Tradeoff Θ High State, Low Radius High Radius, Low State A generic cut-and-shift procedure for implementing the Radius -> tradeoff: have r = 4 want r = #slots = (2r + )/(2k + ) to go from radius r to radius k.

26 The Radius/State Tradeoff Let s apply radius/state tradeoff cut-and-shift procedure. grailed To get a make the trade, the algorithm adds two extra slots.

27 The Radius/State Tradeoff WinGless grailed SLoPpy Mid HedgeHog Cubitus Interruptus repressor Cubitus Interruptus (protein) Cubitus Interruptus (mrna) Focus on the three proteins: Sloppy, Wingless, Engrailed. SLoPpy WinGless EnGrailed Interesting superficial resemblance:......

28 The Radius/State Tradeoff Much stronger form of validation comes by isolating a path within the gene regulatory network that stabilizes the three proteins in a spatial feedback sequence: en wg en wg en wg wg slp 2 HH hh CN ci 2 ci CN HH hh 2 wg en mid Mid slp en wg en mid Mid ci hh HH 2 CN wg ci 3 mid CN Mid ci HH hh 2 ci 3 The dynamically activated subnetworks: Mid slp hh en CN HH Mid CN 2 3

29 Mid slp hh en CN HH Mid CN Consolidating the intermediates: Conclusion: the coordination of multiple input-output state relations that unfold over space and time allow cellular agents with nearest-neighbor views to generate long-range coordinate patterns. Experimental: look in related drosophilids for modifications (DePace lab)

30 Beyond Drosophila (kinda, sorta, maybe)

31 What does thinking of natural spatial multi-agent systems qua computers tell us scientically? It tells about local information requres, helping understand: What agent resource capacities are required to solve a given task. But the things we learned (i.e. cut and shift) were general. Can we then turn around this knowledge to build a better bug? so that we could, for example, make general patterns? In theory.

32 Designing Local Rules Goal: Design local rules to make given patterns. Repeat patterns have a well-defined gradient. Θ (B) + = { i, if B i is consistent with Θ B(2R), otherwise For example: -repeat pattern (which has an r = 2 check scheme), Θ () + = Θ () + = Otherwise, Θ () + = Θ () + = Θ (b) = b(3)

33 Designing Local Rules Now simply define: Θ Θ F (B) = + (B( : 2r)) Θ Θ Θ 2r(Θ) B( : 2R) + Θ Θ Θ Θ Gradient spreads correctness. Θ Gradient waves... Multiple waves all at once in actuality... Not hard to implement in higher dimensions.

34 Designing Local Rules Resulting rule is % robust to: Initial condition perturbations. Timing issues.

35 Designing Local Rules But: the rule to solve a repeat pattern of size q takes a radius of q. Can run a radius-minimization algorithm. And then, couple in the radius/state tradeoff. A) B) Anterior Posterior Anterior Posterior

36 Designing Local Rules Similar principles can be applied to develop a language for building a diverse array of patterns.

37 Designing Local Rules Pattern Local representation Θ Θ min Minimize radius Tradeoff Dynamic Rule F( ) Network???

38 Designing Local Rules Some VERY simple ideas in multi-cellular synthetic biology might be ``understandable (or at any rate, imagineable) and connectable to cellular programs Is anything here doable? even remotely? What is the simplest system that could possibly exploit some of these ideas? What I d really like to explore is... evolution.